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In this paper we continue our recent development [1] of the theory of fixed point theorems of nonlinear operators whose domain and range are different Banach spaces. In particular we consider the analogues of recent results of Caristi and Kirk [5,6,8] where "inwardness conditions" are used to obtain fixed points. More precisely "inwardness conditions" on a mapping T whose domain K is a proper subspace of its range have been imposed to ensure that T maps points x of K "towards" K. Caristi and Kirk, for example, have considered two different conditions, metrically inward and weakly inward (this is the tangential boundary condition used in studying, for example, differential equations on closed sets [9]). These conditions are much weaker than the simple inwardness condition that T map the boundary of K into K.